Pattern avoidance and quasisymmetric functions
| Autor: | Zachary Hamaker, Brendan Pawlowski, Bruce E. Sagan |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Mathematics::Combinatorics
Conjecture Generating function Sigma Combinatorics Symmetric group Homogeneous space FOS: Mathematics Pi Mathematics - Combinatorics Discrete Mathematics and Combinatorics 05E05 (Primary) 05A05 (Secondary) Combinatorics (math.CO) Element (category theory) Descent (mathematics) Mathematics |
| Zdroj: | Algebraic Combinatorics. 3:365-388 |
| ISSN: | 2589-5486 |
| DOI: | 10.5802/alco.96 |
| Popis: | Given a set of permutations Pi, let S_n(Pi) denote the set of permutations in the symmetric group S_n that avoid every element of Pi in the sense of pattern avoidance. Given a subset S of {1,...,n-1}, let F_S be the fundamental quasisymmetric function indexed by S. Our object of study is the generating function Q_n(Pi) = sum F_{Des sigma} where the sum is over all sigma in S_n(Pi) and Des sigma is the descent set of sigma. We characterize those Pi contained in S_3 such that Q_n(Pi) is symmetric or Schur nonnegative for all n. In the process, we show how each of the resulting Pi can be obtained from a theorem or conjecture involving more general sets of patterns. In particular, we prove results concerning symmetries, shuffles, and Knuth classes, as well as pointing out a relationship with the arc permutations of Elizalde and Roichman. Various conjectures and questions are mentioned throughout. Comment: 3 figures, 28 pages, added some questions |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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